Prove the identity (1/sinx - 1/tanx)^2 -= (1-cosx)/(1+cosx). ?

Prove the identity (1/sinx - 1/tanx)^2 -= (1-cosx)/(1+cosx).

1 Answer
Mar 13, 2018

I would start with the left hand side, by rewriting in terms of sine and cosine.

LHS:

(1/sinx - 1/(sinx/cosx))^2

(1/sinx - cosx/sinx)^2

((1 - cosx)/sinx)^2

(1 -cosx)^2/sin^2x

Recall that

sin^2x +cos^2x = 1 -> sin^2x= 1- cos^2x.

(1 - cosx)^2/(1 - cos^2x)

Now do a little factoring.

((1 - cosx)(1 - cosx))/((1 + cosx)(1 - cosx))

(1 - cosx)/(1 + cosx)

We now see that LHS = RHS, therefore we've proven this identity.

Hopefully this helps!